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Modeling of a metrological AFM interferometric position measurement system to determine its measurement uncertainty
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Laboratoire national de métrologie et d’essais
P. Ceriaa, S. Ducourtieuxa, Y. Boukellala, N. Feltina,
A. Allardb and N. Fischer b
a LNE, Nanometrology
b LNE, Mathematics and Statistics Department
Modeling of a metrological AFM
interferometric position
measurement system to determine
its measurement uncertainty
JRP 6DoF : metrology for movement and
positioning in six degrees of freedom
Content
1. Introduction
2. The principal features of the model
3. The used statistical tools
4. The link with the experimental data
5. The most significant results
6. Conclusion and perspectives
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Introduction
Why do we develop a model of the LNE’s metrological AFM ?
The development of the instrument started in 2007. We have now reached the
final phase where we have to deal with the evaluation of the uncertainty budget for
the positioning of the instrument. But the task is not so easy when the specific
metrology loop of the instrument is considered !
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Sebastien Ducourtieux and Benoit Poyet, Meas. Sci. Technol. 22 (2011) 094010
Introduction
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x
y
z
The metrology loop is composed of 4 differential interferometers (Renishaw), 2
zerodur prisms (one linked to the tip, the other one to the sample) and 8 mirrors.
Introduction
Original configuration for the interferometers with very good performances :
- low noise : 0,3 nm at 1,5 MHz
- stability estimated to only few nanometers over one hour.
BUT it is difficult to feel how much sensitive the system is to factors like :
- Parasitic rotations, interferometer misalignment (Abbe error)
- Cosine error
- Non perfect geometry of the prisms (angles of the mirror, shapes and
roughness of the mirror)
- Positioning error
- Basic dilatations
-
=> This is the reason why we came to a virtual representation of the metrology
system in order to better estimate the uncertainty budget of the instrument and
also to gain knowledge on its behavior.
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Principal features of the model
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How do we model the metrology system ?
The model is developed under Matlab using object oriented programming.
- Mirrors are modeled by a 3D cloud of points
1) plane mirror
2) filters to produce
random or predefined
shape and roughness on
the mirror
3) translation and rotation of the
mirror using homogeneous
coordinates formalism
Principal features of the model
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The prism is simply created by placing each mirror at its correct position
according to mAFM CAD drawing. Each prism is then considered as an object
which can be also translated and rotated using the homogeneous coordinates
formalism.
The interferometers are considered as a black box which only measures the
distance separating the interferometer head and the mirror. Each beam possesses
a source point, a direction vector and a Gaussian lateral extension. The relative
displacement is evaluated according to the following formulas :
Principal features of the model
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distance between the beam axis
and the considered point on the
mirror
Principal features of the model
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A set of about 100 parameters are used to control the model and define the
geometry of the metrology system. Some of these parameters are then coupled
by using other parameters or laws to define some specific behavior (i.e.
homothetic dilatation of the prism or structural frame). Each parameter is
associated with a distribution law and an uncertainty according to our actual
knowledge and our experimental data.
We feed the model with experimental data
Link with experimental data
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Flatness and roughness measurement
using interferometric microscope*
Angular relations between the prism mirror
using the LNE angular platen and
using theodolite system*
*Performed by the Optic Group team from Soleil Synchrotron
Beam intensity profile
Stage Parasitic rotations
Instrument and room thermal stability
Model
Principal features of the model
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GUI
We use Monte Carlo method to propagate the probability density function
associated to each parameter inside the model and then to evaluate the position
measurement uncertainty.
Implemented statistical tools
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10 min for 105 draws and
100 parameters on the
cluster
Quick overview of the results
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Quick overview of the results
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XYZ positioning uncertainty with
the actual set of parameters (not
the definitive one) and for the full
range of displacement
(60 x 60 x 15 µm3):
about 6 nm
Morri’s plan is used to identify the most influential components by evaluating their
interactions two by two :
- Influential due to many interactions with other components (increasing )
- Influential because of the strength of the effect (increasing )
Quick overview of the results
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Mean effect
Standard deviation of the effect
More effect
More interaction
Quick overview of the results
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Mean effect
Standard deviation of the effect
More effect
More interaction
Most influential parameters with the actual set of parameters (not the definitive
one) and for the full range of displacement (60 x 60 x 15 µm3) :
- Parasitic rotations (high mean effect and high interaction)
- Abbe offset (high mean effect and high interaction)
- Sample thickness (high mean effect and high interaction)
- Orthogonally error (high mean effect but no interaction)
Morri’s plan is used to identify the most influential components by evaluating their
interactions two by two :
- Influential due to many interactions with other components (increasing )
- Influential because of the strength of the effect (increasing )
Sobol’ indices are used evaluate the contribution of each parameter to the global
uncertainty :
Quick overview of the results
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Identification of the contribution of each parameter to the global uncertainty using
Sobol’ indices :
Quick overview of the results
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Contribution to the global uncertainty in the actual configuration :
- 75 % coming from Abbe error (parasitic rotations, Abbe Offset and sample
height)
- 24 % coming from the orthogonality error
Conclusion
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- Uncertainty is mainly impacted by Abbe error and orthogonality error
=> for orthogonality, we can introduce corrections
=> for Abbe error, we must improve beam alignment and if possible, bring
some modifications to the stage to reduce parasitic rotations.
- In the actual configuration with the actual set of parameters, the reached
positioning uncertainty for the LNE’s mAFM is 6 nm for the full range of
displacement (60 x 60 x 15 µm3) , but the definitive experimental data are not
used here so it will be reduced !
- Easily reconfigurable model that we can adapted to another configuration or
another instrument (for example to evaluate instruments implementing 6 DoF
displacement and/or 6 DoF metrology loop).
Perspective
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Now, we are working on the modeling of the image construction.
We add a modeled grating and a modeled AFM tip
Objective : to move from the positioning uncertainty to the measurement
uncertainty of standard dimensional properties (i.e. pitch and step height).
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Thank you for your attention
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Thank you for your attention
Implemented statistical tools
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